Category:Radius of Convergence

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This category contains results about Radius of Convergence.


Let $\xi \in \C$ be a complex number.

For $z \in \C$, let:

$\displaystyle f \paren z = \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n$

be a power series about $\xi$.


The radius of convergence is the extended real number $R \in \overline \R$ defined by:

$R = \displaystyle \inf \set {\cmod {z - \xi}: z \in \C, \sum_{n \mathop = 0}^\infty a_n \paren {z - \xi}^n \text{ is divergent} }$

where a divergent series is a series that is not convergent.

As usual, $\inf \O = +\infty$.